Counting Rational Points on Danielewski and Double Danielewski Surfaces over Finite Fields

Let $\text{Fq}$ be the finite field with $q$ elements. We study the number of $\text{Fq}$-rational points on Danielewski and double Danielewski surfaces. For Danielewski surfaces, the point count is reduced to the number of roots of $P(Z)$ over $\text{Fq}\mathrm{.}$ For double Danielewski surfaces, one has to count the number of tuples $(\text{be},\text{g})\in {\text{Fq}}^2$, such that $P(0,\text{g})=0$, $Q(0,\text{be},\text{g})=0$ hold simultaneously. We compute these numbers using gcd methods, resultants, character sums, Gauss sums, and the König–Rados theorem. We obtain explicit formulas in several structured cases, derive general bounds, and give a Macaulay2 algorithm for verification and show an intresting connection between the number of $\text{Fq}$-rational points of these surfaces and polygonal numbers.